Are you using your own data for the network dataset or are you using our routing service?
If you're using your own data (whether locally or in a service), then on the network dataset, you can set up the Turn category evaluator—ArcGIS Pro | ArcGIS Desktop in Pro or the About global turns—Help | ArcGIS Desktop (Global Turn Delay Evaluator) in ArcMap. Keep in mind that using this will alter the final cost that gets returned for the route (because it'll be adding seconds on to every turn). But it should help you avoid left turns, if you set the left turn up to have a higher delay than the other turns.
Let me know if this helps!
Thanks Rachel, this is a huge help. My goal is to actually assign a cost to the turns. So don't prevent them entirely but add a cost so they are only taken when necessary. I will dig in and take a look. I had no idea this existed.
Yes, we are using our own Network for now but will likely want to purchase StreetMap Premium. Would this not be usable/available via StreetMap Premium?